3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.
y = 2 cos 3x on [−π,π]

Verified step by step guidance

1. Step 1: Recall the formula for the arc length of a curve y = f(x) on the interval [a, b]. The arc length is given by: ${\int }_a^b\sqrt{1+{\frac{\text{d}y}{\text{d}x}}^2}\text{d}x$.
2. Step 2: Compute the derivative of y = 2 cos(3x) with respect to x. Using the chain rule, the derivative is: $\frac{\text{d}y}{\text{d}x}=-6\text{sin}\left(3x\right)$.
3. Step 3: Substitute the derivative into the arc length formula. The integrand becomes: $\sqrt{1+{(-6\text{sin}\left(3x\right))}^2}$.
4. Step 4: Simplify the square of the derivative. Squaring $-6\text{sin}\left(3x\right)$ gives: ${-6\text{sin}\left(3x\right)}^2=36{\text{sin}\left(3x\right)}^2$.
5. Step 5: Write the integral for the arc length. The integral becomes: ${\int }_{-}^{\pi }\sqrt{1+36{\text{sin}\left(3x\right)}^2}\text{d}x$.

Key Concepts

Arc Length Formula

The arc length of a curve defined by a function y = f(x) from x = a to x = b is given by the integral L = ∫[a to b] √(1 + (dy/dx)²) dx. This formula derives from the Pythagorean theorem, where the infinitesimal segments of the curve are approximated as straight lines.

Derivative of a Function

The derivative of a function, denoted as dy/dx, represents the rate of change of the function with respect to x. For the curve y = 2 cos(3x), finding the derivative is essential to apply the arc length formula, as it will be squared and added to 1 under the square root in the integral.

Definite Integral

A definite integral calculates the accumulation of quantities, such as length, over a specified interval [a, b]. In this context, the integral will be set up from -π to π, representing the total arc length of the curve y = 2 cos(3x) over that interval, without evaluating it.